Cryptol Reference Manual
Basic Syntax
Declarations
f x = x + y + z
Type Signatures
f,g : {a,b} (fin a) => [a] b
Layout
Groups of declarations are organized based on indentation. Declarations with the same indentation belong to the same group. Lines of text that are indented more than the beginning of a declaration belong to that declaration, while lines of text that are indented less terminate a group of declarations. Consider, for example, the following Cryptol declarations:
f x = x + y + z
where
y = x * x
z = x + y
g y = y
This group has two declarations, one for f and one for g. All the
lines between f and g that are indented more than f belong to
f. The same principle applies to the declarations in the where block
of f, which defines two more local names, y and z.
Identifiers
Cryptol identifiers consist of one or more characters. The first
character must be either an English letter or underscore (_). The
following characters may be an English letter, a decimal digit,
underscore (_), or a prime ('). Some identifiers have special
meaning in the language, so they may not be used in programmer-defined
names (see Keywords and Built-in Operators).
name name1 name' longer_name
Name Name2 Name'' longerName
Keywords and Built-in Operators
The following identifiers have special meanings in Cryptol, and may not be used for programmer defined names:
else include property let infixl parameter
extern module then import infixr constraint
if newtype type as infix down
private pragma where hiding primitive by
The following table contains Cryptol’s operators and their associativity with lowest precedence operators first, and highest precedence last.
Operator |
Associativity |
|---|---|
|
right |
|
right |
|
right |
|
not associative |
|
not associative |
|
right |
|
left |
|
right |
|
right |
|
left |
|
left |
|
left |
|
right |
|
left |
(unary) |
right |
Built-in Type-level Operators
Cryptol includes a variety of operators that allow computations on the numeric types used to specify the sizes of sequences.
Operator |
Meaning |
|---|---|
|
Addition |
|
Subtraction |
|
Multiplication |
|
Division |
|
Ceiling division ( |
|
Modulus |
|
Ceiling modulus (compute padding) |
|
Exponentiation |
|
Ceiling logarithm (base 2) |
|
Bit width (equal to |
|
Maximum |
|
Minimum |
Numeric Literals
Numeric literals may be written in binary, octal, decimal, or
hexadecimal notation. The base of a literal is determined by its prefix:
0b for binary, 0o for octal, no special prefix for
decimal, and 0x for hexadecimal.
254 // Decimal literal
0254 // Decimal literal
0b11111110 // Binary literal
0o376 // Octal literal
0xFE // Hexadecimal literal
0xfe // Hexadecimal literal
Numeric literals in binary, octal, or hexadecimal notation result in
bit sequences of a fixed length (i.e., they have type [n] for
some n). The length is determined by the base and the number
of digits in the literal. Decimal literals are overloaded, and so the
type is inferred from context in which the literal is used. Examples:
0b1010 // : [4], 1 * number of digits
0o1234 // : [12], 3 * number of digits
0x1234 // : [16], 4 * number of digits
10 // : {a}. (Literal 10 a) => a
// a = Integer or [n] where n >= width 10
Numeric literals may also be written as polynomials by writing a polynomial
expression in terms of x between an opening <| and a closing |>.
Numeric literals in polynomial notation result in bit sequences of length
one more than the degree of the polynomial. Examples:
<| x^^6 + x^^4 + x^^2 + x^^1 + 1 |> // : [7], equal to 0b1010111
<| x^^4 + x^^3 + x |> // : [5], equal to 0b11010
Cryptol also supports fractional literals using binary (prefix 0b),
octal (prefix 0o), decimal (no prefix), and hexadecimal (prefix ox)
digits. A fractional literal must contain a . and may optionally
have an exponent. The base of the exponent for binary, octal,
and hexadecimal literals is 2 and the exponent is marked using the symbol p.
Decimal fractional literals use exponent base 10, and the symbol e.
Examples:
10.2
10.2e3 // 10.2 * 10^3
0x30.1 // 3 * 64 + 1/16
0x30.1p4 // (3 * 64 + 1/16) * 2^4
All fractional literals are overloaded and may be used with types that support
fractional numbers (e.g., Rational, and the Float family of types).
Some types (e.g. the Float family) cannot represent all fractional literals
precisely. Such literals are rejected statically when using binary, octal,
or hexadecimal notation. When using decimal notation, the literal is rounded
to the closest representable even number.
All numeric literals may also include _, which has no effect on the
literal value but may be used to improve readability. Here are some examples:
0b_0000_0010
0x_FFFF_FFEA
Expressions
This section provides an overview of the Cryptol’s expression syntax.
Calling Functions
f 2 // call `f` with parameter `2`
g x y // call `g` with two parameters: `x` and `y`
h (x,y) // call `h` with one parameter, the pair `(x,y)`
Prefix Operators
-2 // call unary `-` with parameter `2`
- 2 // call unary `-` with parameter `2`
f (-2) // call `f` with one argument: `-2`, parens are important
-f 2 // call unary `-` with parameter `f 2`
- f 2 // call unary `-` with parameter `f 2`
Infix Functions
2 + 3 // call `+` with two parameters: `2` and `3`
2 + 3 * 5 // call `+` with two parameters: `2` and `3 * 5`
(+) 2 3 // call `+` with two parameters: `2` and `3`
f 2 + g 3 // call `+` with two parameters: `f 2` and `g 3`
- 2 + - 3 // call `+` with two parameters: `-2` and `-3`
- f 2 + - g 3
Type Annotations
Explicit type annotations may be added on expressions, patterns, and in argument definitions.
x : Bit // specify that `x` has type `Bit`
f x : Bit // specify that `f x` has type `Bit`
- f x : [8] // specify that `- f x` has type `[8]`
2 + 3 : [8] // specify that `2 + 3` has type `[8]`
\x -> x : [8] // type annotation is on `x`, not the function
if x
then y
else z : Bit // the type annotation is on `z`, not the whole `if`
[1..9 : [8]] // specify that elements in `[1..9]` have type `[8]`
f (x : [8]) = x + 1 // type annotation on patterns
Todo
Patterns with type variables
Explicit Type Instantiation
If f is a polymorphic value with type:
f : { tyParam } tyParam
f = zero
you can evaluate f, passing it a type parameter:
f `{ tyParam = 13 }
Local Declarations
Local declarations have the weakest precedence of all expressions.
2 + x : [T]
where
type T = 8
x = 2 // `T` and `x` are in scope of `2 + x : `[T]`
if x then 1 else 2
where x = 2 // `x` is in scope in the whole `if`
\y -> x + y
where x = 2 // `y` is not in scope in the defintion of `x`
Block Arguments
When used as the last argument to a function call,
if and lambda expressions do not need parens:
f \x -> x // call `f` with one argument `x -> x`
2 + if x
then y
else z // call `+` with two arguments: `2` and `if ...`
Conditionals
The if ... then ... else construct can be used with
multiple branches. For example:
x = if y % 2 == 0 then 22 else 33
x = if y % 2 == 0 then 1
| y % 3 == 0 then 2
| y % 5 == 0 then 3
else 7
Demoting Numeric Types to Values
The value corresponding to a numeric type may be accessed using the following notation:
`t
Here t should be a finite type expression with numeric kind. The resulting expression will be of a numeric base type, which is sufficiently large to accommodate the value of the type:
`t : {a} (Literal t a) => a
This backtick notation is syntax sugar for an application of the number primtive, so the above may be written as:
number`{t} : {a} (Literal t a) => a
If a type cannot be inferred from context, a suitable type will be automatically chosen if possible, usually Integer.
Basic Types
Bits
The type Bit has two inhabitants: True and False. These values
may be combined using various logical operators, or constructed as
results of comparisons.
Operator |
Associativity |
Description |
|
right |
Short-cut implication |
|
right |
Short-cut or |
|
right |
Short-cut and |
!= == |
none |
Not equals, equals |
|
none |
Comparisons; $ indicates signed |
|| |
right |
Logical or |
^ |
left |
Exclusive-or |
&& |
right |
Logical and |
~ |
right |
Logical negation |
Tuples and Records
Tuples and records are used for packaging multiple values together. Tuples are enclosed in parentheses, while records are enclosed in curly braces. The components of both tuples and records are separated by commas. The components of tuples are expressions, while the components of records are a label and a value separated by an equal sign. Examples:
(1,2,3) // A tuple with 3 component
() // A tuple with no components
{ x = 1, y = 2 } // A record with two fields, `x` and `y`
{} // A record with no fields
The components of tuples are identified by position, while the components of records are identified by their label, and so the ordering of record components is not important for most purposes. Examples:
(1,2) == (1,2) // True
(1,2) == (2,1) // False
{ x = 1, y = 2 } == { x = 1, y = 2 } // True
{ x = 1, y = 2 } == { y = 2, x = 1 } // True
Ordering on tuples and records is defined lexicographically. Tuple components are compared in the order they appear, whereas record fields are compared in alphabetical order of field names.
Accessing Fields
The components of a record or a tuple may be accessed in two ways: via pattern matching or by using explicit component selectors. Explicit component selectors are written as follows:
(15, 20).0 == 15
(15, 20).1 == 20
{ x = 15, y = 20 }.x == 15
Explicit record selectors may be used only if the program contains sufficient type information to determine the shape of the tuple or record. For example:
type T = { sign : Bit, number : [15] }
// Valid definition:
// the type of the record is known.
isPositive : T -> Bit
isPositive x = x.sign
// Invalid definition:
// insufficient type information.
badDef x = x.f
The components of a tuple or a record may also be accessed using pattern matching. Patterns for tuples and records mirror the syntax for constructing values: tuple patterns use parentheses, while record patterns use braces. Examples:
getFst (x,_) = x
distance2 { x = xPos, y = yPos } = xPos ^^ 2 + yPos ^^ 2
f p = x + y where
(x, y) = p
Selectors are also lifted through sequence and function types, point-wise, so that the following equations should hold:
xs.l == [ x.l | x <- xs ] // sequences
f.l == \x -> (f x).l // functions
Thus, if we have a sequence of tuples, xs, then we can quickly obtain a
sequence with only the tuples’ first components by writing xs.0.
Similarly, if we have a function, f, that computes a tuple of results,
then we can write f.0 to get a function that computes only the first
entry in the tuple.
This behavior is quite handy when examining complex data at the REPL.
Updating Fields
The components of a record or a tuple may be updated using the following notation:
// Example values
r = { x = 15, y = 20 } // a record
t = (True,True) // a tuple
n = { pt = r, size = 100 } // nested record
// Setting fields
{ r | x = 30 } == { x = 30, y = 20 }
{ t | 0 = False } == (False,True)
// Update relative to the old value
{ r | x -> x + 5 } == { x = 20, y = 20 }
// Update a nested field
{ n | pt.x = 10 } == { pt = { x = 10, y = 20 }, size = 100 }
{ n | pt.x -> x + 10 } == { pt = { x = 25, y = 20 }, size = 100 }
Sequences
A sequence is a fixed-length collection of elements of the same type.
The type of a finite sequence of length n, with elements of type a
is [n] a. Often, a finite sequence of bits, [n] Bit, is called a
word. We may abbreviate the type [n] Bit as [n]. An infinite
sequence with elements of type a has type [inf] a, and [inf] is
an infinite stream of bits.
[e1,e2,e3] // A sequence with three elements
[t1 .. t2] // Enumeration
[t1 .. <t2] // Enumeration (exclusive bound)
[t1 .. t2 by n] // Enumeration (stride)
[t1 .. <t2 by n] // Enumeration (stride, ex. bound)
[t1 .. t2 down by n] // Enumeration (downward stride)
[t1 .. >t2 down by n] // Enumeration (downward stride, ex. bound)
[t1, t2 .. t3] // Enumeration (step by t2 - t1)
[e1 ...] // Infinite sequence starting at e1
[e1, e2 ...] // Infinite sequence stepping by e2-e1
[ e | p11 <- e11, p12 <- e12 // Sequence comprehensions
| p21 <- e21, p22 <- e22 ]
x = generate (\i -> e) // Sequence from generating function
x @ i = e // Sequence with index binding
arr @ i @ j = e // Two-dimensional sequence
Note: the bounds in finite sequences (those with ..) are type expressions, while the bounds in infinite sequences are value expressions.
Operator |
Description |
|---|---|
|
Sequence concatenation |
|
Shift (right, left) |
|
Rotate (right, left) |
|
Arithmetic right shift (on bitvectors only) |
|
Access elements (front, back) |
|
Access sub-sequence (front, back) |
|
Update the value of a sequence at a location (front, back) |
|
Update multiple values of a sequence (front, back) |
There are also lifted pointwise operations.
[p1, p2, p3, p4] // Sequence pattern
p1 # p2 // Split sequence pattern
Functions
\p1 p2 -> e // Lambda expression
f p1 p2 = e // Function definition
Type Declarations
Type Synonyms
type T a b = [a] b
A type declaration creates a synonym for a
pre-existing type expression, which may optionally have
arguments. A type synonym is transparently unfolded at
use sites and is treated as though the user had instead
written the body of the type synonym in line.
Type synonyms may mention other synonyms, but it is not
allowed to create a recursive collection of type synonyms.
Newtypes
newtype NewT a b = { seq : [a]b }
A newtype declaration declares a new named type which is defined by
a record body. Unlike type synonyms, each named newtype is treated
as a distinct type by the type checker, even if they have the same
bodies. Moreover, types created by a newtype declaration will not be
members of any typeclasses, even if the record defining their body
would be. For the purposes of typechecking, two newtypes are
considered equal only if all their arguments are equal, even if the
arguments do not appear in the body of the newtype, or are otherwise
irrelevant. Just like type synonyms, newtypes are not allowed to form
recursive groups.
Every newtype declaration brings into scope a new function with the
same name as the type which can be used to create values of the
newtype.
x : NewT 3 Integer
x = NewT { seq = [1,2,3] }
Just as with records, field projections can be used directly on values of newtypes to extract the values in the body of the type.
> sum x.seq
6
Modules
A module is used to group some related definitions. Each file may contain at most one top-level module.
module M where
type T = [8]
f : [8]
f = 10
Hierarchical Module Names
Module may have either simple or hierarchical names.
Hierarchical names are constructed by gluing together ordinary
identifiers using the symbol ::.
module Hash::SHA256 where
sha256 = ...
The structure in the name may be used to group together related
modules. Also, the Cryptol implementation uses the structure of the
name to locate the file containing the definition of the module.
For example, when searching for module Hash::SHA256, Cryptol
will look for a file named SHA256.cry in a directory called
Hash, contained in one of the directories specified by CRYPTOLPATH.
Module Imports
To use the definitions from one module in another module, we use
import declarations:
// Provide some definitions
module M where
f : [8]
f = 2
// Uses definitions from `M`
module N where
import M // import all definitions from `M`
g = f // `f` was imported from `M`
Import Lists
Sometimes, we may want to import only some of the definitions from a module. To do so, we use an import declaration with an import list.
module M where
f = 0x02
g = 0x03
h = 0x04
module N where
import M(f,g) // Imports only `f` and `g`, but not `h`
x = f + g
Using explicit import lists helps reduce name collisions. It also tends to make code easier to understand, because it makes it easy to see the source of definitions.
Hiding Imports
Sometimes a module may provide many definitions, and we want to use most of them but with a few exceptions (e.g., because those would result to a name clash). In such situations it is convenient to use a hiding import:
module M where
f = 0x02
g = 0x03
h = 0x04
module N where
import M hiding (h) // Import everything but `h`
x = f + g
Qualified Module Imports
Another way to avoid name collisions is by using a qualified import.
module M where
f : [8]
f = 2
module N where
import M as P
g = P::f
// `f` was imported from `M`
// but when used it needs to be prefixed by the qualifier `P`
Qualified imports make it possible to work with definitions that happen to have the same name but are defined in different modules.
Qualified imports may be combined with import lists or hiding clauses:
import A as B (f) // introduces B::f
import X as Y hiding (f) // introduces everything but `f` from X
// using the prefix `X`
It is also possible to use the same qualifier prefix for imports from different modules. For example:
import A as B
import X as B
Such declarations will introduces all definitions from A and X
but to use them, you would have to qualify using the prefix B::.
Private Blocks
In some cases, definitions in a module might use helper functions that are not intended to be used outside the module. It is good practice to place such declarations in private blocks:
module M where
f : [8]
f = 0x01 + helper1 + helper2
private
helper1 : [8]
helper1 = 2
helper2 : [8]
helper2 = 3
The private block only needs to be indented if it might be followed by additional public declarations. If all remaining declarations are to be private then no additional indentation is needed as the private block will extend to the end of the module.
module M where
f : [8]
f = 0x01 + helper1 + helper2
private
helper1 : [8]
helper1 = 2
helper2 : [8]
helper2 = 3
The keyword private introduces a new layout scope, and all declarations
in the block are considered to be private to the module. A single module
may contain multiple private blocks. For example, the following module
is equivalent to the previous one:
module M where
f : [8]
f = 0x01 + helper1 + helper2
private
helper1 : [8]
helper1 = 2
private
helper2 : [8]
helper2 = 3
Parameterized Modules
Warning
This section documents the current design, but we are in the process of redesigning some aspects of the parameterized modules mechanism.
module M where
parameter
type n : # // `n` is a numeric type parameter
type constraint (fin n, n >= 1)
// Assumptions about the parameter
x : [n] // A value parameter
// This definition uses the parameters.
f : [n]
f = 1 + x
Named Module Instantiations
One way to use a parameterized module is through a named instantiation:
// A parameterized module
module M where
parameter
type n : #
x : [n]
y : [n]
f : [n]
f = x + y
// A module instantiation
module N = M where
type n = 32
x = 11
y = helper
helper = 12
The second module, N, is computed by instantiating the parameterized
module M. Module N will provide the exact same definitions as M,
except that the parameters will be replaced by the values in the body
of N. In this example, N provides just a single definition, f.
Note that the only purpose of the body of N (the declarations
after the where keyword) is to define the parameters for M.
Parameterized Instantiations
It is possible for a module instantiation to be itself parameterized. This could be useful if we need to define some of a module’s parameters but not others.
// A parameterized module
module M where
parameter
type n : #
x : [n]
y : [n]
f : [n]
f = x + y
// A parameterized instantiation
module N = M where
parameter
x : [32]
type n = 32
y = helper
helper = 12
In this case N has a single parameter x. The result of instantiating
N would result in instantiating M using the value of x and 12
for the value of y.
Importing Parameterized Modules
It is also possible to import a parameterized module without using a module instantiation:
module M where
parameter
x : [8]
y : [8]
f : [8]
f = x + y
module N where
import `M
g = f { x = 2, y = 3 }
A backtick at the start of the name of an imported module indicates that we are importing a parameterized module. In this case, Cryptol will import all definitions from the module as usual, however every definition will have some additional parameters corresponding to the parameters of a module. All value parameters are grouped in a record.
This is why in the example f is applied to a record of values,
even though its definition in M does not look like a function.
Comments
Cryptol supports block comments, which start with
/*and end with*/, and line comments, which start with//and terminate at the end of the line. Block comments may be nested arbitrarily.Todo
Document
/** */